U32.and_low_hx: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend as U32
Lines 221-227 of bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend&start_line=221&end_line=227

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as SC
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S

def and_low_hx(+m: Nat, +d: Nat, +xb: Bool, +xt: Word(m), +hx: {Nat.is_lt(Nat.add(S.bit_value(xb), Nat.double(S.unsigned(m, xt))), SC.pow2(d)) == True{} : Bool}) -> {Nat.is_lt(S.unsigned(m, xt), SC.pow2(and_low_prev(d))) == True{} : Bool}:
  match d:
    case 0n:
      %Equal.sym(Nat, S.unsigned(m, xt), 0n, N.bit_split_val(xb, False{}, S.unsigned(m, xt), 0n, lt_one_zero(Nat.add(S.bit_value(xb), Nat.double(S.unsigned(m, xt))), hx))) : {Nat.is_lt(_, 1n) == True{} : Bool}
      {==}
    case 1n+k:
      N.bit_double_lt(xb, S.unsigned(m, xt), SC.pow2(k), hx)

Uses (14):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=unsigned
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=bit_double_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=bit_split_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend&name=and_low_prev
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend&name=lt_one_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=pow2